Question

Help please! :) Find the coefficient k such that.... T=(kZ)/sqrt(X^2)has a t distribution with 8 degrees freedom....Pasting the entire question into the comments..... THANK YOU in advance!

Answer 1

The \(T\) distribution can be defined as the following ratio between independent standard normal distribution \(Z\) and chi-squared distribution \(\chi^2\) with \(n\) degrees of freedom:
\[T=\frac{Z}{\sqrt{\dfrac{\chi^2}{n}}}\]
\(T\) will have the same degrees of freedom \(n\). You want to find \(k\) such that the \(n\) above is \(8\). Some algebraic manipulation will clear things up:
\[T=\frac{Z}{\sqrt{\dfrac{\chi^2}{n}}}=\frac{\color{red}{\sqrt n} Z}{\sqrt{\chi^2}}=\frac{\color{red}kZ}{\sqrt{\chi^2}}\]

Answer 2

Thank you!